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Fitting models to data – IV (Yet more on Maximum Likelihood Estimation)

Fitting models to data – IV (Yet more on Maximum Likelihood Estimation). Fish 458, Lecture 11. The Poisson Distribution. The density function : rt is the expected number of events ( r is a rate and t is time). k is the number of discrete events (count data).

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Fitting models to data – IV (Yet more on Maximum Likelihood Estimation)

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  1. Fitting models to data – IV(Yet more on Maximum Likelihood Estimation) Fish 458, Lecture 11

  2. The Poisson Distribution • The density function: • rt is the expected number of events (r is a rate and t is time). • k is the number of discrete events (count data). • The Poisson distribution has only one parameter (rt) which is both the mean and the variance. However, often we find the variance is larger than would be expected under the Poisson model so assume this model with care – better still, look at the negative binomial distribution first!

  3. Poisson Model – Example-I 100 longline sets are observed and the following data are collected. What is the rate (numbers per set) at which seabird are captured?

  4. Poisson Model – Example-II • The log-likelihood function (after removal of constants) is given by: • This equation is maximized at r=0.69. • How else could we have obtained the same estimate for r?

  5. The Negative Binomial Distribution-I • The negative binomial distribution extends the Poisson distribution by allowing the rate parameter to be a (gamma) random variable: • R is the expected number of observations (discrete or continuous) • k is an “overdispersion” parameter. • Note:

  6. The Negative Binomial Distribution-II • The mean of the negative binomial distribution is R. • The variance of the negative binomial distribution is: • The negative binomial distribution collapses to the Poisson distribution as

  7. The Negative Binomial Distribution-III

  8. The Negative Binomial Distribution-IV • Consider the case in which we monitor the catch of a given species (in number) as a function of fishing effort. If the catch occurs randomly per unit time we would expect the catch to be Poisson distributed with mean (and variance) equal to the product of the fishing duration and a rate of capture. • For this problem, we apply the Poisson model and the Negative binomial model.

  9. The Negative Binomial Distribution-V k is 84 for this fit – good evidence that the Poisson model is adequate.

  10. The Negative Binomial Distribution-VI The data are now (really) overdispersed relative to a Poisson distribution. The estimates are again identical, but the negative binomial indicates lesser precision than the Poisson.

  11. Overdispersion • Overdisperson implies that the variance of the data is greater than that expected under the distribution assumed (e.g. Poisson  variance=mean). • If the data are overdispersed but this is ignored, you are overweighting the data (i.e. underestimating their uncertainty).

  12. Likelihood “Cheat sheet” Data? Continuous Discrete Number of outcomes Can be negative? 2 Yes No Many Normal / t Binomial lognormal / gamma Poisson / Negative binomial / Multinomial

  13. Fitting – Miscellany-II • Robustness • In many cases, the assumptions underlying the likelihood function are wrong: “some data points are too unlikely”. Such data points are outliers. • Outliers can either be left out of the analysis or the likelihood “robustified” to reduce their influence. • Robustification includes: minimizing the median residual, leaving out the largest residuals, downweighting large residuals.

  14. Fitting – Miscellany-III • Contradictory data • All probability statements are based on the assumptions of the model and likelihood function, and these may be wrong! • Often when we have two (or more) data sources they disagree! • The problem is that (at least) one data source is not measuring what we think it is. • Solutions: • Include some probability that each index doesn’t tell us anything; and • Run separate assessments for each index in turn.

  15. Contradictory data(northern cod) The northern cod dilemma: two abundance indices – one increasing (and relatively precise), the other not (and noisy). To pool or not to pool!

  16. Additional Readings • Chen, Y. Fournier, D. 1999. Can. J. Fish. Aquat. Sci. 56: 1525 – 1533. • Fournier, D.A.; Hampton, J.; Sibert, J.R. 1998. Can. J. Fish. Aquat. Sci. 55: 2105–2116. • Schnute, J.T.; Hilborn, R. 1993. Can. J. Fish. Aquat. Sci. 50: 1916 – 1927.

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